and we note that \(\ell _u|_{L^2_V} = 0\) .
From this, one uses an adaptation of de Rham's theorem to invert the linear operator in (REF ).
We refer the reader to Corollary 2.4 of [1]} for a precise statement of de Rham's theorem which roughly states that \(\nabla \colon L^2_0(\Omega (t))\rightarrow \lbrace  g \in H^{-1}(\Omega (t)) : g|_{V(t)} =0 \rbrace \)  is an isomorphism for each \(t \in (0,T)\) .
